3 Exercise You have $10,000 to invest for 5 years. How much additional interest will you earn if the investment offers a 5% annual interest, compared to a 4% annual interest? What is the annual interest rate if the $10,000 grows into $40,000 in 20 years? How long will it take the $10,000 to double in value if it earns 5% each year?

4 Objectives for Today Determine future and present values of multiple cash flows Understand annuities and perpetuities Learn how to compute loan payments Learn how to find the interest rate on a loan Understand how interest rates are quoted

5 Future Value of Multiple Cash Flows Suppose you deposit $100 today in an account that pays 8% per year. In one year, you will deposit another $100. How much will you have in two years?

6 Future Value of Multiple Cash Flows (continued) Another way to approach the problem: FV of $100 deposited in Year 0 = FV of $100 deposited in Year 1 = Total =

7 Example: Multiple Cash Flows over 5 Years Consider the future value of $2,000 invested at the end of each of the next 5 years: The Rolling Forward Method

8 Example: Multiple Cash Flows over 5 Years The Adding Up Method Note: Unless otherwise specified, always assume that cash flows occur at the end of each period.

9 Exercise If you are investing $1,000, $2,000, and $3,000 in one, two and three years, respectively, in an investment account that pays 8% per year. How much money will you have in 5 years?

10 Present Value of Multiple Cash Flows Suppose you need $1,000 in one year and $2,000 more in two years. If you can earn 9% on your money, how much do you have to put up today to cover these amounts in the future? In other words, what is the present value of the two cash flows at 9%? Year $1,000 $2,000

11 Present Value of Multiple Cash Flows (continued) Suppose that you have an investment that pays $1,000 at the end of every year for the next 5 years and that the current interest rate is 6%. The Rolling Backward Method

13 Exercise Investment X offers to pay you $3,000 per year for 5 years, where Investment Y offers to pay you $5,000 per year for 3 years. If the discount rate is 8%, which one of the investments should you choose?

14 Special Cases of Multiple Cash Flows Annuities An annuity is a series of fixed cash flows that occur at the end of each period for some fixed number of periods Examples: Perpetuities A perpetuity is a special case of an annuity when the cash flows continues forever Example:

15 Present Value for Annuities Suppose you are examining an investment that pays $600 at the end of each of the next 3 years. What is the present value of this annuity if the discount rate is 12%? What if the investment pays $5 at the end of each month for the next 30 years? Present Value Interest Factor for Annuities PV (annuity) = C x = C x 1-1 PVIF r 1 (1 +r ) t r

16 Present Value for Annuities (continued) Suppose you are examining an investment that pays $600 at the end of each of the next 3 years. What is the present value of this annuity if the discount rate is 12%? PV (annuity) = $600 x 1-1 (1 +12% ) 3 = $1,441 12% You can also use an annuity table: PVIFA(12%, 3) = What if the investment pays $5 at the end of each month for the next 30 years? Assume that the monthly discount rate is 1%.

17 Exercise Congratulations! You just won the $250 million Powerball lottery. You can choose between the following 2 options: a single cash payment of $120 million today; or an annual payment of $10 million over the next 25 years, with the first payment to be received today Assume the discount rate is 10%. Which option should you choose?

18 Annuities: Finding the Payment Suppose you want to borrow $20,000 to buy a new car. You can borrow at 12% per year, compounded monthly (i.e. 1% per month). If you take a 5-year loan with Bank of America, what will your monthly payment be?

19 Annuities: Finding the Number of Payments Suppose you want to borrow $20,000 to buy a new car. You can borrow at 12% per year, compounded monthly (i.e. 1% per month). If you are going to make a monthly payment of $500, how long will it take before you pay off the loan?

20 Annuities: Finding the Interest Rate Suppose you borrow the $20,000 from your parents. You agree to pay them back $400 every month for 5 years. What is the implicit interest rate you are paying?

21 Future Value for Annuities You plan to contribute $1,000 every year into a retirement account that pays 8% per year. If you retire in 50 years, how much money will you have? Future Value Interest Factor for Annuities FV (annuity) = C x = C x FVIF 1 r (1 + r) t 1 r

22 Present vs. Future Values for Annuities Consider that you invest $1,000 every year for 50 years in a retirement account that pays 8% per year. FV(annuity) = $573,770 PV(annuity) = $1,000 x PVIFA(8%, 50) = $12,233 If we compound the present value of the annuity, $12,233, at 8% per year for 50 years, we have: $12,233 x (1 + 8%) 50 = $573,770

23 Annuities Due Ordinary Annuities Cash flows occur at the end of each period, e.g. first loan payment usually happens one month after you get the loan Usually the case, unless otherwise specified Annuities Due Cash flows occur at the beginning of each period, e.g. lease, insurance Example: an annuity due has 5 payments of $400 each and the relevant discount rate is 10%

24 Perpetuities Present Value PV(perpetuity) = C r Example: Suppose that you own an investment that offers a perpetual cash flow of $500 every year. The relevant discount rate is 10%. What is the value of the investment? PV(perpetuity) = Future Value $500 10% = $5,000

26 Effective Annual Rate Steps to Compute Effective Annual Rate: Divide the quoted rate by the number of times the interest is compounded Add 1 and raise to the power of the number of times the interest is compounded Subtract 1 EAR = (1 + Quoted Rate/m) m -1 EAR Bank A: 10% compounded semiannually Bank B: 10% compounded quarterly Bank C: 9% compounded monthly

27 Annual Percentage Rate Interest rate charged per period multiplied by the number of periods per year Example: A typical credit card charges an interest rate of 24% APR with monthly payments. What is the actual interest rate you pay on such a credit card?

28 Exercise National PayDay allows you to write a check for $125 dated one month in the future, for which they give you $100 today. What are the APR and EAR for this arrangement?

29 Recap Future and present values of multiple cash flows Future and present values for annuities and perpetuities Annuity: a series of constant cash flows for a fixed number of periods Perpetuity: an annuity that lasts forever Computation of loan payments, number of payments, and interest rate How interest rates are quoted: EAR vs. APR

30 Homework You have arranged a 5-year auto loan of $30,000 with Wells Fargo to buy a new car. Your monthly payment will be $625. What is the APR of the loan? What is the EAR?

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